Distributed Abstract Optimization via Constraints Consensus: Theory and Applications

Distributed abstract programs are a novel class of distributed optimization problems where (i) the number of variables is much smaller than the number of constraints and (ii) each constraint is associated to a network node. Abstract optimization programs are a generalization of linear programs that captures numerous geometric optimization problems. We propose novel constraints consensus algorithms for distributed abstract programs: as each node iteratively identifies locally active constraints and exchanges them with its neighbors, the network computes the active constraints determining the global optimum. The proposed algorithms are appropriate for networks with weak time-dependent connectivity requirements and tight memory constraints. We show how suitable target localization and formation control problems can be tackled via constraints consensus.

💡 Research Summary

The paper introduces a novel class of distributed optimization problems called “distributed abstract programs,” in which the number of constraints far exceeds the number of decision variables (or the combinatorial dimension δ). An abstract optimization program (also known as an LP‑type or abstract linear program) is defined by a finite constraint set H and a value function φ that satisfies monotonicity and locality. The optimal solution is characterized by a minimal subset of constraints, called a basis, whose cardinality is at most δ. Classical geometric problems such as the smallest enclosing ball, stripe, or annulus fit this framework, as do ordinary linear programs (δ = d).

The authors propose “constraints‑consensus” algorithms to solve such problems in a network of processors where each node holds a single constraint. Each node maintains a local basis B and repeatedly performs two primitive operations: (i) a violation test Viol(B,h) to check whether a newly received constraint h would improve the current solution, and (ii) a basis update Basis(B,h) that recomputes a minimal basis for B∪{h} when a violation occurs. The updated basis is then broadcast to neighboring nodes. This exchange continues until all nodes hold the same global basis B*; at that point the algorithm terminates.

Three algorithmic variants are presented: (1) a nominal version that stores the full set of active constraints and includes a distributed halting condition; (2) a memory‑saving version that keeps only the current basis (size O(δ)) and discards older constraints; (3) a version robust to time‑varying directed graphs, requiring only that the sequence of communication digraphs be jointly strongly connected over bounded windows. The paper proves monotonicity of the basis size, finite‑time convergence to consensus, and exact convergence to the unique optimal solution when the abstract program is basis‑regular. A distributed stopping rule based on local counters and the absence of basis changes is also derived, eliminating the need for a global synchronizer.

Complexity analysis yields a conservative upper bound on the number of communication rounds; the authors conjecture a linear dependence on the number of constraints n. To support this claim, they conduct extensive Monte‑Carlo simulations on two families of randomly generated linear programs and three network topologies (line, Erdős‑Rényi, random geometric). Across tens of thousands of trials, the average convergence time scales roughly linearly with n, and the distributed algorithm matches the solution quality of centralized LP solvers while using only O(δ) memory per node.

Two concrete applications illustrate the practical relevance of the framework.

1. Target Localization: Sensors each generate half‑plane constraints describing the feasible region of a moving target. An “eight‑half‑plane consensus” algorithm computes an axis‑aligned bounding box that encloses the target. By embedding the consensus step into a set‑membership filter (prediction‑update recursion), the method tracks the target’s region in real time with modest communication overhead (only eight constraints exchanged per round).
2. Formation Control: Mobile robots aim to achieve geometric formations (point, line, circle). Each robot exchanges constraints that encode reachable shapes in minimum time, runs the consensus algorithm to agree on a common shape, and simultaneously moves toward that shape while preserving network connectivity via a standard connectivity‑maintenance controller. In the limit of infinitesimal motion per communication step, the “move‑to‑consensus‑shape” strategy solves the optimal formation problem. Simulations with up to 20 robots show rapid convergence to the desired formation and robustness to topology changes.

The authors contrast their approach with prior work on distributed SVM training and parallel LP solvers, highlighting that earlier methods either let the stored constraint set grow to O(n) or required static, fully connected networks. In contrast, constraints‑consensus maintains O(δ) memory, tolerates time‑varying graphs, and provides a provable distributed halting condition.

In summary, the paper demonstrates that abstract optimization—through the lens of LP‑type problems—can be efficiently solved in highly constrained, dynamic distributed settings by means of a simple yet powerful constraints‑consensus mechanism. The theoretical results, extensive Monte‑Carlo validation, and concrete applications to sensor‑based localization and multi‑robot formation control collectively establish a versatile foundation for future research on distributed geometric and machine‑learning optimization under severe resource limitations.